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Author: Sajan K. Samuel Publisher: ISBN: Category : Differential equations, Elliptic Languages : en Pages : 0
Book Description
"One of the most important and useful tools used in the study of partial differential equations is the maximum principle. This principle is a natural extension to higher dimensions of an elementary fact of calculus: any function, which satisfies the inequality f′′ > 0 on an interval [a,b], achieves its maximum at one of the endpoints of the interval. In this context, we say that the solution to the differential inequality f′′ > 0 satisfies a maximum principle. In this thesis we will discuss the maximum principles for partial differential equations and their applications. More precisely, we will show how one may employ the maximum principles to obtain information about uniqueness, approximation, boundedness, convexity, symmetry or asymptotic behavior of solutions, without any explicit knowledge of the solutions themselves. The thesis will be organized in two main parts. The purpose of the first part is to briefly introduce in Chapter 1 the terminology and the main tools to be used throughout this thesis. We will start by introducing the second order linear differential operators of elliptic and parabolic type. Then, we will develop the first and second maximum principles of E. Hopf for elliptic equations, respectively the maximum principles of L. Nirenberg and A. Friedman for parabolic equations. Next, in the second part, namely in Chapter 2 and 3, we will introduce various P-functions, which are nothing else than appropriate functional combinations of the solutions and their derivatives, and derive new maximum principles for such functionals. Moreover, we will show how to employ these new maximum principles to get isoperimetric inequalities, symmetry results and convexity results in the elliptic case (Chapter 2), respectively spatial and temporal asymptotic behavior of solutions, in the parabolic case (Chapter 3)."--Abstract.
Author: Sajan K. Samuel Publisher: ISBN: Category : Differential equations, Elliptic Languages : en Pages : 0
Book Description
"One of the most important and useful tools used in the study of partial differential equations is the maximum principle. This principle is a natural extension to higher dimensions of an elementary fact of calculus: any function, which satisfies the inequality f′′ > 0 on an interval [a,b], achieves its maximum at one of the endpoints of the interval. In this context, we say that the solution to the differential inequality f′′ > 0 satisfies a maximum principle. In this thesis we will discuss the maximum principles for partial differential equations and their applications. More precisely, we will show how one may employ the maximum principles to obtain information about uniqueness, approximation, boundedness, convexity, symmetry or asymptotic behavior of solutions, without any explicit knowledge of the solutions themselves. The thesis will be organized in two main parts. The purpose of the first part is to briefly introduce in Chapter 1 the terminology and the main tools to be used throughout this thesis. We will start by introducing the second order linear differential operators of elliptic and parabolic type. Then, we will develop the first and second maximum principles of E. Hopf for elliptic equations, respectively the maximum principles of L. Nirenberg and A. Friedman for parabolic equations. Next, in the second part, namely in Chapter 2 and 3, we will introduce various P-functions, which are nothing else than appropriate functional combinations of the solutions and their derivatives, and derive new maximum principles for such functionals. Moreover, we will show how to employ these new maximum principles to get isoperimetric inequalities, symmetry results and convexity results in the elliptic case (Chapter 2), respectively spatial and temporal asymptotic behavior of solutions, in the parabolic case (Chapter 3)."--Abstract.
Author: Murray H. Protter Publisher: Springer Science & Business Media ISBN: 1461252822 Category : Mathematics Languages : en Pages : 271
Book Description
Maximum Principles are central to the theory and applications of second-order partial differential equations and systems. This self-contained text establishes the fundamental principles and provides a variety of applications.
Author: Gershon Kresin Publisher: American Mathematical Soc. ISBN: 0821889818 Category : Mathematics Languages : en Pages : 330
Book Description
The main goal of this book is to present results pertaining to various versions of the maximum principle for elliptic and parabolic systems of arbitrary order. In particular, the authors present necessary and sufficient conditions for validity of the classical maximum modulus principles for systems of second order and obtain sharp constants in inequalities of Miranda-Agmon type and in many other inequalities of a similar nature. Somewhat related to this topic are explicit formulas for the norms and the essential norms of boundary integral operators. The proofs are based on a unified approach using, on one hand, representations of the norms of matrix-valued integral operators whose target spaces are linear and finite dimensional, and, on the other hand, on solving certain finite dimensional optimization problems. This book reflects results obtained by the authors, and can be useful to research mathematicians and graduate students interested in partial differential equations.
Author: Emmanuel Franck Publisher: Springer Nature ISBN: 3031408640 Category : Mathematics Languages : en Pages : 381
Book Description
This volume comprises the first part of the proceedings of the 10th International Conference on Finite Volumes for Complex Applications, FVCA, held in Strasbourg, France, during October 30 to November 3, 2023. The Finite Volume method, and several of its variants, is a spatial discretization technique for partial differential equations based on the fundamental physical principle of conservation. Recent decades have brought significant success in the theoretical understanding of the method. Many finite volume methods are also built to preserve some properties of the continuous equations, including maximum principles, dissipativity, monotone decay of the free energy, asymptotic stability, or stationary solutions. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete level. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. In recent years, the efficient implementation of these methods in numerical software packages, more specifically to be used in supercomputers, has drawn some attention. This volume contains all invited papers, as well as the contributed papers focusing on finite volume schemes for elliptic and parabolic problems. They include structure-preserving schemes, convergence proofs, and error estimates for problems governed by elliptic and parabolic partial differential equations. The second volume is focused on finite volume methods for hyperbolic and related problems, such as methods compatible with the low Mach number limit or able to exactly preserve steady solutions, the development and analysis of high order methods, or the discretization of kinetic equations.
Author: Catherine Bandle Publisher: Springer Science & Business Media ISBN: 3764373849 Category : Mathematics Languages : en Pages : 466
Book Description
Haim Brezis has made significant contributions in the fields of partial differential equations and functional analysis, and this volume collects contributions by his former students and collaborators in honor of his 60th anniversary at a conference in Gaeta. It presents new developments in the theory of partial differential equations with emphasis on elliptic and parabolic problems.
Author: Nikolaĭ Vladimirovich Krylov Publisher: American Mathematical Soc. ISBN: 082180569X Category : Mathematics Languages : en Pages : 178
Book Description
These lectures concentrate on fundamentals of the modern theory of linear elliptic and parabolic equations in H older spaces. Krylov shows that this theory - including some issues of the theory of nonlinear equations - is based on some general and extremely powerful ideas and some simple computations. The main object of study is the first boundary-value problems for elliptic and parabolic equations, with some guidelines concerning other boundary-value problems such as the Neumann or oblique derivative problems or problems involving higher-order elliptic operators acting on the boundary. Numerical approximations are also discussed. This book, containing 200 exercises, aims to provide a good understanding of what kind of results are available and what kinds of techniques are used to obtain them.
Author: Prof. Dr. Pavol Quittner Publisher: Springer ISBN: 3030182223 Category : Mathematics Languages : en Pages : 719
Book Description
This book is devoted to the qualitative study of solutions of superlinear elliptic and parabolic partial differential equations and systems. This class of problems contains, in particular, a number of reaction-diffusion systems which arise in various mathematical models, especially in chemistry, physics and biology. The first two chapters introduce to the field and enable the reader to get acquainted with the main ideas by studying simple model problems, respectively of elliptic and parabolic type. The subsequent three chapters are devoted to problems with more complex structure; namely, elliptic and parabolic systems, equations with gradient depending nonlinearities, and nonlocal equations. They include many developments which reflect several aspects of current research. Although the techniques introduced in the first two chapters provide efficient tools to attack some aspects of these problems, they often display new phenomena and specifically different behaviors, whose study requires new ideas. Many open problems are mentioned and commented. The book is self-contained and up-to-date, it has a high didactic quality. It is devoted to problems that are intensively studied but have not been treated so far in depth in the book literature. The intended audience includes graduate and postgraduate students and researchers working in the field of partial differential equations and applied mathematics. The first edition of this book has become one of the standard references in the field. This second edition provides a revised text and contains a number of updates reflecting significant recent advances that have appeared in this growing field since the first edition.
Author: Clément Cancès Publisher: Springer ISBN: 3319573942 Category : Mathematics Languages : en Pages : 530
Book Description
This book is the second volume of proceedings of the 8th conference on "Finite Volumes for Complex Applications" (Lille, June 2017). It includes reviewed contributions reporting successful applications in the fields of fluid dynamics, computational geosciences, structural analysis, nuclear physics, semiconductor theory and other topics. The finite volume method in its various forms is a space discretization technique for partial differential equations based on the fundamental physical principle of conservation, and recent decades have brought significant advances in the theoretical understanding of the method. Many finite volume methods preserve further qualitative or asymptotic properties, including maximum principles, dissipativity, monotone decay of free energy, and asymptotic stability. Due to these properties, finite volume methods belong to the wider class of compatible discretization methods, which preserve qualitative properties of continuous problems at the discrete l evel. This structural approach to the discretization of partial differential equations becomes particularly important for multiphysics and multiscale applications. The book is useful for researchers, PhD and master’s level students in numerical analysis, scientific computing and related fields such as partial differential equations, as well as for engineers working in numerical modeling and simulations.
Author: Patrizia Pucci Publisher: Springer Science & Business Media ISBN: 3764381450 Category : Mathematics Languages : en Pages : 240
Book Description
Maximum principles are bedrock results in the theory of second order elliptic equations. This principle, simple enough in essence, lends itself to a quite remarkable number of subtle uses when combined appropriately with other notions. Intended for a wide audience, the book provides a clear and comprehensive explanation of the various maximum principles available in elliptic theory, from their beginning for linear equations to recent work on nonlinear and singular equations.
Author: Sajan K. Samuel
Author: Sajan K. Samuel
Author: Murray H. Protter
Author: Sperb
Author: Gershon Kresin
Author: Emmanuel Franck
Author: Catherine Bandle
Author: Nikolaĭ Vladimirovich Krylov
Author: Prof. Dr. Pavol Quittner
Author: Clément Cancès
Author: Patrizia Pucci